Definitions and Evolutions of Statistical Entropy for Hamiltonian Systems

Abstract: Regardless of studies and debates over a century, the statistical origin of the second law of thermodynamics still remains illusive. One essential obstacle is the lack of a proper theoretical formalism for non-equilibrium entropy. Here I revisit the seminal ideas about non-equilibrium statistical entropy due to Boltzmann and due to Gibbs, and synthesize them into a coherent and precise framework. Using this framework, I clarify the anthropomorphic principle of entropy, and analyze the evolution of entropy for classical Hamiltonian systems under different experimental setups. I find that evolution of Boltzmann entropy obeys a Stochastic H-Theorem, which relates probability of Boltzmann entropy increasing to that of decreasing. By contrast, the coarse-grained Gibbs entropy is monotonically increasing, if the microscopic dynamics is locally mixing, and the initial state is a Boltzmann state. These results clarify the precise meaning of the second law of thermodynamics for classical systems, and demonstrate that it is the initial condition as a Boltzmann state that is ultimately responsible for the arrow of time.